Latent factor models reduce the dimensionality of data by converting a large number of discrete or continuous observed variables (called items) into a smaller number of continuous unobserved variables (called latent factors), potentially making the data easier to understand. Latent factor models for discrete items are called item response theory (IRT) models.

Traditional maximum likelihood (ML) estimation methods for IRT models are computationally intensive when the sample size, the number of items, and the number of latent factors are all large. This issue can be avoided by approximating the ML estimator using an importance-weighted amortized variational estimator (I-WAVE) from the field of deep learning (for details, see Urban and Bauer, 2021). As an estimation byproduct, I-WAVE allows researchers to compute approximate factor scores and log-likelihoods for any observation — even new observations that were not used for model fitting.

DeepIRTools' main functionality is the stand-alone IWAVE class contained in the iwave module. This class includes fit(), scores(), and log_likelihood() methods for fitting a latent factor model and for computing approximate factor scores and log-likelihoods for the fitted model.

The following (multidimensional) latent factor models are currently available...

• ...for binary and ordinal items:
  • Graded response model
  • Generalized partial credit model
  • Nominal response model
• ...for continuous items:
  • Normal (linear) factor model
  • Lognormal factor model
• ...for count data:
  • Poisson factor model
  • Negative binomial factor model

DeepIRTools supports mixing item types, handling missing completely at random data, and predicting the mean of the latent factors with covariates (i.e., latent regression modeling); all models are estimable in both confirmatory and exploratory contexts. In the confirmatory context, constraints on the factor loadings, intercepts, and factor covariance matrix are implemented by providing appropriate arguments to fit(). In the exploratory context, the screeplot() function in the figures module may help identify the number of latent factors underlying the data.

• Python 3.7 or higher
• torch
• pyro-ppl
• numpy

To install the latest version:

pip install deepirtools

Official documentation is available here.

See big_5_tutorial.ipynb for a tutorial on using DeepIRTools to fit several kinds of latent factor models using large-scale data.

See mnist_demo.ipynb for a demonstration of how DeepIRTools may be used to fit a flexible and identifiable model for generating realistic handwritten digits.

In [1]: import deepirtools
   ...: from deepirtools import IWAVE
   ...: import torch

In [2]: deepirtools.manual_seed(123)

In [3]: data = deepirtools.load_grm()["data"]

In [4]: n_items = data.shape[1]

In [5]: model = IWAVE(
   ...:       model_type = "grm",
   ...:       latent_size = 4,
   ...:       n_cats = [3] * n_items,
   ...:       Q = torch.block_diag(*[torch.ones([3, 1])] * 4),
   ...:       correlated_factors = [0, 1, 2, 3],
   ...: )

Initializing model parameters
Initialization ended in  0.0  seconds

In [6]: model.fit(data, iw_samples = 10)

Fitting started
Epoch =      100 Iter. =    25201 Cur. loss =   10.68   Intervals no change =  100
Fitting ended in  109.23  seconds

In [7]: model.loadings # Loadings matrix.
Out[7]: 
tensor([[0.8295, 0.0000, 0.0000, 0.0000],
        [0.5793, 0.0000, 0.0000, 0.0000],
        [0.7116, 0.0000, 0.0000, 0.0000],
        [0.0000, 0.7005, 0.0000, 0.0000],
        [0.0000, 1.1687, 0.0000, 0.0000],
        [0.0000, 1.2890, 0.0000, 0.0000],
        [0.0000, 0.0000, 0.9268, 0.0000],
        [0.0000, 0.0000, 1.2653, 0.0000],
        [0.0000, 0.0000, 1.5622, 0.0000],
        [0.0000, 0.0000, 0.0000, 1.0346],
        [0.0000, 0.0000, 0.0000, 1.3641],
        [0.0000, 0.0000, 0.0000, 1.1348]])

In [8]: model.intercepts # Category intercepts.
Out[8]: 
tensor([[ 2.4245, -0.1637],
        [ 1.8219, -1.0013],
        [ 2.0811, -1.1320],
        [ 0.0948, -1.7253],
        [ 2.6597, -2.3412],
        [ 0.2610, -1.4938],
        [ 2.8196, -1.3281],
        [ 0.4833, -2.8053],
        [ 1.6395, -2.2220],
        [ 1.3482, -1.8870],
        [ 2.1606, -2.8600],
        [ 2.5318, -0.1333]])

In [9]: model.cov # Factor covariance matrix.
Out[9]: 
tensor([[ 1.0000, -0.0737,  0.2130,  0.2993],
        [-0.0737,  1.0000, -0.1206, -0.3031],
        [ 0.2130, -0.1206,  1.0000,  0.1190],
        [ 0.2993, -0.3031,  0.1190,  1.0000]])
        
In [10]: model.log_likelihood(data) # Approximate log-likelihood.

Computing approx. LL
Approx. LL computed in 33.27 seconds
Out[6]: -85961.69088745117

In [11]: model.scores(data) # Approximate factor scores.
Out[11]: 
tensor([[-0.6211,  0.1301, -0.7207, -0.7485],
        [ 0.2189, -0.2649,  0.0109, -0.2363],
        [ 0.0544,  0.9308,  0.7940, -0.8851],
        ...,
        [-0.2964, -0.9597, -0.8885, -0.0057],
        [-1.6015,  0.9812,  0.0486,  0.1773],
        [ 2.0448,  0.0583,  1.2005, -0.9317]])

To cite DeepIRTools in publications, use:

• Urban, C. J., & He, S. (2022). DeepIRTools: Deep learning-based estimation and inference for item response theory models. Python package. https://github.com/cjurban/deepirtools

To cite the method, use:

• Urban, C. J., & Bauer, D. J. (2021). A deep learning algorithm for high-dimensional exploratory item factor analysis. Psychometrika, 86(1), 1-29. https://link.springer.com/article/10.1007/s11336-021-09748-3

and/or:

• Urban, C. J. (2021). Machine learning-based estimation and goodness-of-fit for large-scale confirmatory item factor analysis (Publication No. 28772217) [Master's thesis, University of North Carolina at Chapel Hill]. ProQuest Dissertations Publishing. https://www.proquest.com/docview/2618877227/21C6C467D6194C1DPQ/

BibTeX entries for LaTeX users are:

@Manual{DeepIRTools,
title = {{D}eep{IRT}ools: {D}eep learning-based estimation and inference for item response theory models},
     author = {Urban, Christopher J. and He, Shara},
     year = {2022},
     note = {Python package},
     url = {https://github.com/cjurban/deepirtools},
}

@article{UrbanBauer2021,
    author = {Urban, Christopher J. and Bauer, Daniel J.},
    year={2021},
    title={{A} deep learning algorithm for high-dimensional exploratory item factor analysis},
    journal = {Psychometrika},
    volume = {86},
    number = {1},
    pages = {1--29}
}

@phdthesis{Urban2021,
    author  = {Urban, Christopher J.},
    title   = {{M}achine learning-based estimation and goodness-of-fit for large-scale confirmatory item factor analysis},
    publisher = {ProQuest Dissertations Publishing},
    school  = {University of North Carolina at Chapel Hill},
    year    = {2021},
    type    = {Master's thesis},
}